Question

The density of material at any point in a solid cylinder, y^2 + z^2 ≤ a^2, 0 ≤ x ≤ b is proportional to the distance of the point from the x-axis. Find the density. Use k as the proportionality constant.
δ(y, z) =
δ(r) =

Answer 1

I think I got the density

Answer 2

\[k \sqrt{y^2+z^2}\]

Answer 3

what do you think @wio ?

Answer 4

It would help if you draw it.

Answer 5

But I believe that your density function is correct

Answer 6

how would I change it to polar coordinates? @wio

Answer 7

\[
r^2 = y^2+z^2
\]

Answer 8

\[
y=r\sin\theta,\, z=r\cos\theta
\]

Answer 9

And \(x\) remains the same as the height component

Answer 10

Since it is a cylinder \(0\leq \theta\lt 2\pi\).
Since \(x\) acts as the height, it can remain the same.

Answer 11

thanks @wio :)

Answer 12

wait so I don't haven an x on my density funtion @wio

Answer 13

\[k \sqrt{rsin \theta+rcos \theta} \]

Answer 14

No, you don't, because the \(x\) value doesn't change the density

Answer 15

Changing the \(x\) value doesn't change the distance from the \(x\) axis

Answer 16

\[
\delta (y,z) = k\sqrt{y^2+z^2} = k\sqrt{r^2} = kr
\]

Answer 17

Well \(k\sqrt{r^2} = k|r| = kr\), we can say \(|r|=r\) because by design \(r\geq0\).

Answer 18

I don't like how they have \(\delta (y,z)\) and \(\delta (r)\). They are two different functions. I suppose that is me nitpicking though.

Answer 19

For example \(\delta(r,r) = k\sqrt{r^2+r^2} = kr\sqrt 2\neq kr=\delta(r)\).

Answer 20

for the second part of this question they ask to set up an integral

Answer 21

\[\int\limits\limits_{?}^{?}\int\limits\limits_{?}^{?}\int\limits\limits_{?}^{?}f(x,r,\theta)dx dr d \theta \]

Answer 22

so it would be \[\int\limits_{0}^{2\pi}\int\limits_{0}^{6}\int\limits_{0}^{5}kr dx d r d \theta \]

Answer 23

cylinder is radius 6 and length 5 @wio

Answer 24

Remember the Jacobian

Answer 25

\[
dydz =rdrd\theta
\]

Answer 26

what's the jacobian?

Answer 27

It's the scaling factor of the differentials

Answer 28

so it would be \[dx r dr d \theta \]

Answer 29

what whould the k value be?

Answer 30

proportinality constant

Answer 31

they ask for exact answer by having k I can not find an exact answer

Answer 32

You can get an exact answer in terms of \(k\)

Answer 33

can k just be x

Answer 34

No, because that wouldn't work

Answer 35

\(k\) is constant with respect to \(x,y,z\)

Answer 36

i'm so confuse I'm sorry